Absolute stability and absolute hyperbolicity in systems with discrete time-delays

Serhiy Yanchuk, Matthias Wolfrum, Tiago Pereira, Dmitry Turaev

Research output: Contribution to journalArticlepeer-review


An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.

Original languageEnglish
Pages (from-to)323-343
Number of pages21
JournalJournal of Differential Equations
StatePublished - 5 May 2022
Externally publishedYes


  • Absolute hyperbolicity
  • Absolute stability
  • Delay differential equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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